A famous result by our mathemagician Persi Diaconis: 7 shuffles is enough. Put it in everyday words, to randomize one set of Poker cards, you need to (Riffle) shuffle at least 7 times. *Side note: according to this lecture notes, the interpretation is not correct. To completely randomize the deck, you need 11 or more. … Continue reading 7 Shuffles is Enough→

For fixed integers $latex r > 0$, and odd $latex g$, a Moore graph is an $latex r$-regular graph of girth $latex g$ which has the minimum number of vertices $latex n$ among all such graphs with the same regularity and girth.

(Recall, A the girth of a graph is the length of its shortest cycle, and it’s regular if all its vertices have the same degree)

Problem (Hoffman-Singleton): Find a useful constraint on the relationship between $latex n$ and $latex r$ for Moore graphs of girth $latex 5$ and degree $latex r$.

Note: Excluding trivial Moore graphs with girth $latex g=3$ and degree $latex r=2$, there are only two known Moore graphs: (a) the Petersen graph and (b) this crazy graph:

The solution to the problem shows that there are only a few cases left to check.

Solution: It is easy to show that the minimum number of vertices of a…

It was a ghostful Monday--a Monday full of non-measurable ghosts. You've probably heard of this one sphere cut into five pieces becomes two spheres story. This Monday Professor Persi Diaconis gave a public Halloween lecture about these ghosts. I would love to share an amazing proof that shows 0.5=0.99 or 0.999 or 0.9999. Well, with … Continue reading Non-Measuable Ghost: Persi’s Halloween Talk→

I used Arnold's book for my first ODE class, and it was a blast. I later on learned more about this mathematician and find his philosophy of doing math is really inspiring. Here is a biography of this mathematician I found online. Written in Chinese. P.S. I'm engrossed by Arnold's way of doing math (geometry, … Continue reading Arnold传略+Arnold’s Article on Teaching Math→

From Don Knuth‘s Theory Lunch Talk at Stanford on Oct 6. I found a blog post talking about the exact same thing, so won’t repeat. The idea behind the proof of this rather combinatorial theorem is interesting. The main idea is to construct a nice mapping between a permutation of (1,2, …., n) and a pair of (finite) sets: (A, B) based on the elements inside the sets.

You can try to figure out and prove when the equality holds by yourself as an exercise.

When is this Theorem useful? For example, in making representative families <– so that we have better memory performance. Think about it!

Too lazy D:< Plzzzz let me know if you made a nice drawing for the proof of the theorem (Shouldn’t be too hard) so that I can include your image here rather than this textbook cover.

The following theorem of Bollobas is an important result from extremal set theory. It has apparently been rediscovered several times by others. It implies certain classical results, such as Sperner’s Theorem, and has some interesting extensions.

Proof. Let $latex X$ be the union of all the $latex A_i$’s and $latex B_i$’s. Consider an arbitrary permutation $latex pi$ of $latex X$. We claim that there is at most one pair $latex (A_i, B_i)$ such that all of the elements of $latex A_i$ precede those of $latex B_i$ in the ordering $latex pi$. Suppose to the contrary that there are two…